pspatreg {pspatreg}

pspatreg: A package to estimate and make inference for spatial and spatio-temporal econometric regression models

Description

pspatreg offers the user a collection of functions to estimate and make inference of geoadditive spatial or spatio-temporal semiparametric regression models of type ps-sim, ps-sar, ps-sem, ps-sarar, ps-sdm, ps-sdem or ps-slx. These type of specifications are very general and they can include parametric and non-parametric covariates, spatial or spatio-temporal non-parametric trends and spatial lags of the dependent and independent variables and/or the noise of the model. The non-parametric terms (either trends or covariates) are modeled using P-Splines. The non-parametric trend can be decomposed in an ANOVA way including main and interactions effects of 2nd and 3rd order. The estimation method can be restricted maximum likelihood (REML) or maximum likelihood (ML).

Details

Some functionalities that have been included in pspatreg package are:

1. Estimation of the semiparametric regression model

pspatreg allows the estimation of geoadditive spatial or spatio-temporal semiparametric regression models which could include:

• An spatial or spatio-temporal trend, that is, a geoadditive model either for cross-section data or for panel data. This trend can be decomposed in main and interaction functions in an ANOVA way. The spatial (or spatio-temporal) trend gather the potential spatial heterogeneity of the data.

• Parametric covariates as usual in regression models.

• Non-parametric covariates in which the functional relationship is estimated from the data. Both the trends and non-parametric covariates are modelled using P-splines.

• Spatial dependence adding spatial lags of the dependent and independent variables as usual in spatial econometric models. These models gather the potential spatial spillovers.

Once specified, the whole model can be estimated using either restricted maximum-likelihood (REML) or maximum likelihood (ML). The spatial econometric specifications allowed in pspatreg are the following ones:

• ps-sim: geoadditive semiparametric model without spatial effects (in addition to the spatial or spatio-temporal trend, if it is included).

  y = f(s_1,s_2,\tau_{t}) y + X \beta + + \sum_{i=1}^k g(z_i) + \epsilon

  where:

  • f(s_1,s_2,\tau_t) is a smooth spatio-temporal trend of the spatial coordinates s1,s_2 and of the temporal coordinates \tau_t.

  • X is a matrix including values of parametric covariates.

  • g(z_i) are non-parametric smooth functions of the covariates z_i.

  • \epsilon ~ N(0,R) where R = \sigma^2 I_T if errors are uncorrelated or it follows an AR(1) temporal autoregressive structure for serially correlated errors.

• ps-slx: geoadditive semiparametric model with spatial lags of the regresors (either parametric or non-parametric):

  y = f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i =1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + \epsilon

  where:

  • W_N is the spatial weights matrix.

  • I_T is an identity matrix of order T (T = 1 for pure spatial data).

• ps-sar: geoadditive semiparametric model with spatial lag of the dependent variable

  y = (\rho*W_{N} \otimes I_T) y + f(s_1,s_2,\tau_{t}) + X \beta + \sum_{i =1}^k g(z_i) + \epsilon

• ps-sem: geoadditive semiparametric model with a spatial lag of the noise of the model

  y = f(s_1,s_2,\tau_{t}) + X \beta + \sum_{i =1}^k g(z_i) + u

  u = (\delta*W_{N} \otimes I_T) u + \epsilon

• ps-sdm: geoadditive semiparametric model with spatial lags of the endogenous variable and of the regressors (spatial durbin model)

  y = (\rho*W_{N} \otimes I_T) y + f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i = 1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + \epsilon

• ps-sdem: geoadditive semiparametric model with spatial errors and spatial lags of the endogenous variable and of the regressors

  y = f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i = 1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + u

  u = (\delta*W_{N} \otimes I_T) u + \epsilon

• ps-sarar: geoadditive semiparametric model with a spatial lag for: both dependent variable and errors

  y = (\rho*W_{N} \otimes I_T) y + f(s_1,s_2,\tau_{t}) + X \beta + (W_{N} \otimes I_T) X \theta + \sum_{i = 1}^k g(z_i) + \sum_{i = 1}^k g((\gamma_i*W_{N} \otimes I_T) z_i) + u

  u = (\delta*W_{N} \otimes I_T) u + \epsilon

2. Plot of the spatial and spatio-temporal trends

Once estimated the geoadditive semiparametric model, some functions of pspatreg are suited to make plots of the spatial or spatio-temporal trends. These functions, named plot_sp2d and plot_sp3d, can deal either with 'sf' objects or 'dataframe' objects including spatial coordinates (see the examples of the functions). The function plot_sptime allows to examine temporal trends for each spatial unit. Eventually, it is also possible to get the plots on nonparametric covariates using plot_terms.

3. Impacts and spatial spillovers

It is very common in spatial econometrics to evaluate the multiplier impacts that a change in the value of a regressor, in a point in the space, has on the explained variable. The pspatreg package allows the computation and inference of spatial impacts (direct, indirect and total) either for parametric covariates or nonparametric covariates (in the last case, the output are impact functions). The function named impactspar compute the impacts for parametric covariates in the usual way using simulation. On the other hand, the function impactsnopar allows the computation of impact functions for nonparametric covariates. For parametric covariates, the method to compute the impacts is the same than the exposed in LeSage and Page (2009). For nonparametric covariates the method is described in the help of the function impactsnopar. Both impact functions have dedicated methods to get a summary, for the parametric covariates, and plots, for the nonparametric covariates, of the direct, indirect and total impacts.

4. Additional methods

The package pspatreg provides the usual methods to extract information of the fitted models. The methods included are:

• anova: provides tables of fitted 'pspatreg' models including information criteria (AIC and BIC), log-likelihood and degrees of freedom of each fitted model. Also allows to perform LR tests between nested models.

• print method is used to print short tables including the values of beta and spatial coefficients as well as p-values of significance test for each coefficient.

• summary method displays the results of the estimation for spatial and spatio-temporal trends, parametric and nonparametric covariates and spatial parameters.

• coef extractor function of the parametric and spatial coefficientes.

• fitted extractor function of the fitted values.

• logLik extractor function of the log-likelihood.

• residuals extractor function of the residuals.

• vcov extractor function of the covariance matrix of the estimated parameters. The argument bayesian (default = 'TRUE') allows to choose between sandwich (frequentist) or bayesian method to compute the variances and covariances. See Fahrmeir et al. (2021) for details.

Datasets

pspatreg includes a spatio-temporal panel database including observations of unemployment, economic variables and spatial coordinates (centroids) for 103 Italian provinces in the period 1996-2019. This database is provided in RData format and can be loaded using the command data(unemp_it, package = "pspatreg"). The database also includes a W spatial neighborhood matrix of the Italian provinces (computed using queen criterium). Furthermore, a map of Italian provinces is also included as an sf object. This map can be used to plot spatial and spatio-temporal trends estimated for each province. Some examples of spatial and spatio-temporal fitted trends are included in the help of the main function of pspatreg package (see especially ?pspatfit). See Minguez, Basile and Durban (2020) for additional details about this database.
source: Italian National Institute of Statistics (ISTAT) https://www.istat.it

For the spatial pure case, the examples included use the household database ames included in AmesHousing package. See the help of ?AmesHousing::make_ames for an explanation of the variables included in this database. Examples of hedonic models including geoadditive spatial econometric regressions are included in the examples of pspatreg package.

Author(s)

Roman Minguez	[e-mail]
Roberto Basile	[e-mail]
Maria Durban	[e-mail]
Gonzalo Espana-Heredia	[e-mail]

References

• Basile, R.; Durban, M.; Minguez, R.; Montero, J. M.; and Mur, J. (2014). Modeling regional economic dynamics: Spatial dependence, spatial heterogeneity and nonlinearities. Journal of Economic Dynamics and Control, (48), 229-245. <doi:10.1016/j.jedc.2014.06.011>

• Eilers, P. and Marx, B. (1996). Flexible Smoothing with B-Splines and Penalties. Statistical Science, (11), 89-121.

• Eilers, P. and Marx, B. (2021). Practical Smoothing. The Joys of P-Splines. Cambridge University Press.

• Fahrmeir, L.; Kneib, T.; Lang, S.; and Marx, B. (2021). Regression. Models, Methods and Applications (2nd Ed.). Springer.

• Lee, D. and Durban, M. (2011). P-Spline ANOVA Type Interaction Models for Spatio-Temporal Smoothing. Statistical Modelling, (11), 49-69. <doi:10.1177/1471082X1001100104>

• Lee, D. J., Durban, M., and Eilers, P. (2013). Efficient two-dimensional smoothing with P-spline ANOVA mixed models and nested bases. Computational Statistics & Data Analysis, (61), 22-37. <doi:10.1016/j.csda.2012.11.013>

• LeSage, J. and Pace, K. (2009). Introduction to Spatial Econometrics. CRC Press, Boca Raton.

• Minguez, R.; Basile, R. and Durban, M. (2020). An Alternative Semiparametric Model for Spatial Panel Data. Statistical Methods and Applications, (29), 669-708. <doi: 10.1007/s10260-019-00492-8>

• Montero, J., Minguez, R., and Durban, M. (2012). SAR models with nonparametric spatial trends: A P-Spline approach. Estadistica Espanola, (54:177), 89-111.

• Rodriguez-Alvarez, M. X.; Kneib, T.; Durban, M.; Lee, D.J. and Eilers, P. (2015). Fast smoothing parameter separation in multidimensional generalized P-splines: the SAP algorithm. Statistics and Computing 25 (5), 941-957. <doi:10.1007/s11222-014-9464-2>

• Wood, S.N. (2017). Generalized Additive Models. An Introduction with R (second edition). CRC Press, Boca Raton.